Polar Coordinates
Comparison of Polar Coordinates and 2 dimensional Rectangular Coordinates
Coordinate systems provide a means of identifying the location of a point on an axis, a plane or,
in higher dimensional space.
In the 2 dimensional Rectangular coo
COMMON POLAR EQUATIONS
LIN ES T HROU GH T HE ORIGIN have the form = o
The value o gives the angle the line makes with the polar axis (measured counter clockwise).
LIN ES N OT T HROU GH T HE ORIGIN have a variety of forms.
A vertical line x = a in rectangu
Comparison of Cylindrical, Spherical and 3 dimensional Rectangular Coordinates
Coordinate systems provide a means of identifying the location of a point on an axis, a plane or, in
higher dimensional space.
The three dimensional Rectangular (Cartesian) Coo
640:151H Honours Calculus, Midterm Exam #1, Autumn 2009
NAME: Signature: ID #: Instructions: This is a closed book exam. Show answers and arguments in the space provided. You may use the back of the pages also, but indicate clearly any such material that
640:151 Calculus I, Midterm Exam #1, Fall 2011
Department of Mathematics, Rutgers University
NAME (PLEASE PRINT):
SIGNATURE:
ID #:
INSTRUCTOR:
SECTION:
Instructions :
Turn o and put away all mobile phones, computers, iPods, etc.
There is one question in
a)
The expression, x2 + 1 / (x 1) explodes at x = 1 because when the denominator of the
function is set = 0, it is found that x = 1, so b is also 1. As a result, the limit as x
approaches 1 from the left of x2 + 1 / (x 1) approaches either infinity or neg
Continuity
In many cases, you can compute
by plugging a in for x:
For example,
This situation arises often enough that it has a name.
Definition. A function
is continuous at a if
This definition really comprises three things, each of which you need to che
Basic Examples for Finding the Volume of a Solid of Revolution with holes using the Methods of
Washers and Shells
The graph below shows a region in the first quadrant bounded by the line x = 3y and the parabola
.
When the region is revolved about the x-ax
The following is a graph of the first derivative f(x) of a function y = f(x).
y
2
Use this graph of f(x) to answer the
following questions about the graph of
f(x).
y = f(x)
1
-7-6-5-4-32-1
123 4 5 6 7
8
x
-2
a. On what interval(s) is the graph of f(x) con
The following is a graph of the first derivative f(x) of a function y = f(x).
You may assume f(x) is defined for all real numbers.
y
2
Use this graph of f(x) to answer the
following questions about the graph of
f(x).
y = f(x)
1
-7-6-5-4-32-1
123 4 5 6 7
8
Problem 52 Chapter 4 Review page 284:
2nd Edition Calculus Early Transcendentals by Jon Rogawski, W.H. Freeman and Company
x
Graph the function f (x) =
2
(x2 4) 3
Since the denominator would be zero at x = 2 and at x = 2,
the vertical asymptotes are the v
Math 151: Prob 52 Chap 4 Review *1
y
Horiz Asymptote: y = 0
Vert Asymptotes:
x= -2
and x = 2
-2
Decreasing
Concave Down
2
Increasing
C Dwn
x
Decreasing
C UP
Concave Up
_ f ( x ) < 0 _|_f ( x ) > 0_|_f (x ) < 0_
_f ( x ) < 0_ | f (x)<0
| f (x)>0 |_f (x) >0
Calculus I Integration: A Very Short Summary
Denition: Antiderivative
The function F (x) is an antiderivative of the function f (x) on an interval I if F (x) = f (x) for all
x in I.
Notice, a function may have innitely many antiderivatives. For example, t
L4: f(left end point) is the height of the rectangle.
f(x) = x2 on [2,14]; n = 4, equal subintervals with length (14 2)/4 = 3
The Riemann Sum will be the signed areas of the rectangles below.
2
5
8
11
14
Example of Riemann Sums for f(x) = x2 on [2,14] wit
Volume of Solids of Revolution Summary
With Coordinate Axes as the Axis of Revolution
M ET HOD
Rotate about x axis
Rotate about y axis
b
DISKS
d
f (x)
2
dx
a
2
dy
c
b
W ASHERS
g (y )
d
(Ro (x)2 (rI (x)2 dx
(Ro (y )2 (rI (y )2 dy
a
c
d
2
SHELLS
b
y g (y )
Basic Examples for Finding the Volume of a Solid of Revolution without holes using the Methods of
Disks and Shells
The graph below shows a region in the first quadrant bounded
by the coordinate axes and the line 2x + 4y = 12. This may be written as y = 3
Determining which container requires the largest work to pump out liquid.
Containers:
The radiuses of the two cone containers are needed in calculating the work to pump
liquid to ground level. As it is given that the volume of the containers are all equal
To find the area between y = sec(x) and y = tan(x), take the integral of (sec(x) tan(x)
dx from 0 to /2 because sec(x) is larger than tan(x), between the interval of 0 and /2.
This information can be obtained by graphing the functions.
The integral of (se
The function r = 1 - sin can be graphed in the r, plane to help graph the cardioid in the
x, y plane. The graph of the function in the r, plane can be derived by evaluating 1 sin for various s from 0 to 2.
By seeing how 1-sin behaves in the r, plane, the
Math 151, Spring 2009, Review Problems for Exam 2 Your first exam is likely to have problems that do not resemble these review problems. 1. A function y = f (x) is defined implicitly by the equation x2 + 3xy + y 2 = 5. dy in terms of x and y. dx (b) Find
Partial solutions to the Math 151 review problems for Exam 2 These are not complete solutions. They are only intended as a way to check your work. (1) The limit is 4. The LHpital Rule is much more ecient than using long division to o factor (x 1)2 out of
Math 151, Fall 2009, Review Problems for Exam 2 Your second exam is likely to have problems that do not resemble these review problems. Partial answers to these problems will be posted in a few days. (1) Find lim 2x4 3x3 + x2 x + 1 . x1 x4 3x3 + 2x2 + x 1
r y V f Y h c e h c f h c y V T S 2q!WqueX222U# p c Y |2gx o q bup2ucsg5eXbwv2Dx o q g p fY Y p tYh t VaY VaV T S p c g p fY Y Y p tYh t VaY VaV T S e c #2gY dbIp2g5eswbIX2I 5x o q 2gY e x o q g q'2ucXgX2 ce ~ WYVahV g Wqew22#2U e d e x o q 5x o q g g q2
u x x9 w w iu x xiR u X f r d xm d y d f V" RR vf f@ f o Ro | " f f x y u w w u R x w X f@ w X w b u q t79 P xu w u w @sdy w x f f f m 5 vW o o f o "Xg tf 5p vf 5p o o o m &go o f cfw_&f m f of f m &yf m o f cfw_&yf of of l o l o cfw_o f cfw_o f &f l X~
E h W g V ` g E RW R PY g R g Y G D x G fD WV RYY g R PY RW e ` R G xUdbRhSvYUvb`mUdSXxf8X'wdyxXxvRSXISbgvgq6wdD h a R W V R Y R R Stddxxd2bEe "q q bW'lvYx5bxw2thxw2SQjwdD ` p ` gRu RrfVRuf R P H G hE af E tmwAShy SwvpdS)dStS " SSbexStdShStvXY gY aR e cRW
Formula Sheet for Math 151, Final Exam EXPONENTIAL AND LOGARITHMIC FUNCTIONS ex+y = ex ey , ln(xy) = ln x + ln y , ln(ex ) = x , ln ex-y = x y ex , ey (ex )y = exy ln(xy ) = y ln(x) loga x = ln x ln a
= ln x - ln y , ax = e(ln a)x ,
eln x = x ,
TRIGONOMET
Math 151, Fall 2009, Review Problems for the Final Exam Your nal exam is likely to have problems that do not resemble these review problems. You should also look at the review problems for the rst two exams. (1) Find the largest interval [a, b] such that